Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality education elliptic-cohomology enriched fibration foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homology homotopy homotopy-theory homotopy-type-theory index-theory infinity integration integration-theory k-theory kan lie lie-theory limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal-logic model model-category-theory monad monoidal monoidal-category-theory morphism motives motivic-cohomology nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics planar pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
  1. Added to Dedekind cut a short remark on the ¬¬\neg\neg-stability of membership in the lower resp. the upper set of a Dedekind cut.

    • CommentRowNumber2.
    • CommentAuthorTobyBartels
    • CommentTimeJan 27th 2014

    Interesting! I didn't quite follow the last bit of your argument, so I rephrased it. (I also regularized the notation of RR vs UU and finagled a link to stable property.)

    • CommentRowNumber3.
    • CommentAuthorIngoBlechschmidt
    • CommentTimeJan 27th 2014
    • (edited Jan 27th 2014)

    Thanks for catching the typo and streamlining the argument! (For the record, my reasoning was as follows: Since bUb \in U, we have ¬¬(bU)\neg\neg(b \in U). Since ¬¬(bL)\neg\neg(b \in L) and ¬¬\neg\neg distributes over \wedge, we have ¬¬(bLbR)\neg\neg(b \in L \wedge b \in R). Since bLbRb \in L \wedge b \in R \Rightarrow \bot and ¬¬\neg\neg is monotone, we have ¬¬\neg\neg\bot, so \bot.)

    The almost-¬¬\neg\neg-stability can be helpful when proving the equivalence of Dedekind cuts with multi-valued Cauchy “sequences” (i.e. certain maps +P()\mathbb{Q}^+ \to P(\mathbb{Q})). Depending on one’s line of thought, of course, one can otherwise get stuck when trying to show that equivalent multi-valued Cauchy sequences define the same Dedekind cut.

  2. Added a short remark that equality of Dedekind cuts is ¬¬\neg\neg-stable.

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 16th 2018

    Added Project Gutenberg link for Beman’s English translation of Continuity and irrational numbers

    diff, v20, current